#8875550

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,10,8,4)(5,13,14,9)(11,12,15,16)(19,25)(23,26)(27,31)(30,32)(34,39)(36,41)(37,45)(42,48);(2,7)(4,10)(5,14)(11,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(36,41)(37,45)(42,48);(2,7)(3,8)(9,13)(11,15)(18,22)(23,26)(24,28)(30,32)(33,34,38,39)(35,41,43,36)(37,40,45,46)(42,47,48,44);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,39,46)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782746);